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Full text of "Treatise On Analysis Vol-Ii"

256       XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

virtue of (14.1.5) again, independent of the choice of left Haar measure //. The
mapping s\-+AG(s) is called the modulus function on G. If/is any /z-integrable
function, the function xt-*f(xs) is therefore also ^-integrable, and we have

f                       1  f

(14.3.1.1)                    f(xs) d/x(*) = A(s" ')    /(x)

J                                  J

In particular, if A is any /i-integrable subset of G, then As is also ju-integrable,
and

(1 4.3.1 .2)                             n(As) = AfcXA).

(14.3.2)    The mapping s\-+&G(s) is a continuous homomorphism ofG into the
multiplicative group R* .

This. follows immediately from the formula (14.3.1.1) and the lemma
(14.1.5.5).

The group G is said to be unimodular if AG(s) = 1 for all s e G. In this case
there is no distinction to be made between left and right Haar measures, and
we call them simply Haar measures on G.

(14.3.3)    If there exists in G a compact neighborhood V of the neutral element e
which is invariant under all inner automorphisms of G, then G is unimodular.
This is the case, in particular, when G is compact, or discrete, or commutative,

For each SE G we have ji(V) = JJL^^VS) = A0(s()ju(V) by (14.3.1.2) and
(14.1.2.4); hence the result, since /u(V) ^ 0 (14.1.2).

(14.3.4)    If u is any left Haar measure on G, then fi = A""1  IJL. If f is any
H-integrable function on G, then the function x\-^f(x^l)^(x)~ 1 is u-integrable,
and we have

(14.3.4.1)

For each 5- e G we have

5C5XA-1  /i) = (50A-1)  (8(^) = (A^^A-1)  (A^)^) = A"1 - u,

which shows that A""1  ju is a right Haar measure on G. Since fi is also a right
Haar measure, there exists a constant a > 0 such that ji = aA""1  /*. It follows
that n = ^(A"1  //T = aA  ji  a2^ (13.14.5), whence a2 = 1 and thereforeoup